An Exponential Lower Bound of the Size of OBDDs Representing Division

1 Introduction The size of an OBDD [1] largely depends on the variable ordering. It is important to clarify the lower bounds of OBDDs representing arithmetic functions. In this report, we investigate a lower bound of the size of OBDDs representing integer division. More precisely, we focus on each bit of the output of the n-bit / n-bit division, and prove that the size of OBDDs representing the i-th bit is (2 (n0i)=8). We also prove the same lower bound on _-OBDDs [3], ^-OBDDs [3] and 8-OBDDs. To prove this lower bound, we introduce a strong fooling set, which has a more restricted property than a fooling set [4]. An Ordered Binary Decision Diagram (OBDD) is a directed acyclic graph which represents a Boolean function. It has two sink nodes labeled by 0 and 1, called the 0-node and the 1-node. The other nodes are called variable nodes, labeled by input variables. Each variable node has exactly two outgoing edges, called 0-edge and 1-edge. A unique source is called the root node. On every path from the root node to a sink node, each variable appears at most once in the same order. Each node represents a Boolean function. If node v is a sink node, func(v) equals the label. If node v is a variable node, func(v) = x 1 func(0-succ(v)) + x 1 func(1-succ(v)), where x is the label of v, 0-succ(v) (1-succ(v)) is the node pointed by the 0-edge (1-edge) of v. The function represented by an OBDD is the one represented by the root node. Two nodes u and v of same label and representing the same function are called to be equivalent. A node whose 0-edge and 1-edge point to the same node is called a redundant node. Here we consider reduced OBDDs which have node no equivalent nodes and no redundant nodes. The size is the number of nodes in the OBDD.